Ewald Summation Method Research Articles

Long-range electrostatic corrections in multipolar/polarizable QM/MM simulations.

Taking long-range electrostatic effects into account in classical and hybrid quantum mechanics-molecular mechanics (QM/MM) simulations is necessary for an accurate description of the system under study. We have recently developed a method, termed long-range electrostatic corrections (LREC), for monopolar QM/MM calculations. Here, we present an extension of LREC for multipolar/polarizable QM/MM simulations within the LICHEM software package. Reaction barriers and QM-MM interaction energies converge with a LREC cutoff between 20 and 25 Å, in agreement with our previous results. Additionally, the LREC approach for the QM-MM interactions can be smoothly combined with standard shifting or Ewald summation methods in the MM calculations. We recommend the use of QM(LREC)/MM(PME), where the QM region is treated with LREC and the MM region is treated with particle mesh Ewald (PME) summation. This combination is an excellent compromise between simplicity, speed, and accuracy for large QM/MM simulations.

Simulations of Coulomb systems with slab geometry using an efficient 3D Ewald summation method.

We present a new approach to efficiently simulate electrolytes confined between infinite charged walls using a 3d Ewald summation method. The optimal performance is achieved by separating the electrostatic potential produced by the charged walls from the electrostatic potential of electrolyte. The electric field produced by the 3d periodic images of the walls is constant inside the simulation cell, with the field produced by the transverse images of the charged plates canceling out. The non-neutral confined electrolyte in an external potential can be simulated using 3d Ewald summation with a suitable renormalization of the electrostatic energy, to remove a divergence, and a correction that accounts for the conditional convergence of the resulting lattice sum. The new algorithm is at least an order of magnitude more rapid than the usual simulation methods for the slab geometry and can be further sped up by adopting a particle-particle particle-mesh approach.

Solvent Role in Molecular Recognition of Patchouli Extraction Process

Patchouli or Pogostemon cablin Benth is an aromatic plant of importance to the fragrant and cosmetic industries. Its secondary metabolites present interesting pharmacological benefits such as antioxidant and antimutagenic properties. This work is an extended study of the published work in ethanol and water solvents using Ewald summation method and mass spectra characterization of patchouli essential oil extracted with three different polar and non-polar solvents. Ewald summation method has reproduced a better radical distribution function (rdf) intensity in the polar ethanol and water solvents using COMPASS force-field. This work concludes that the complex molecular interaction particularly hydrogen bonding play a significant role to affect the solubility of patchoulol solute either in polar or non-polar solvents during the extraction process.

Molecular Recognition and Solubility of Mefenamic Acid in Water

Molecular dynamics (MD) simulations were performed using COMPASS force field and Ewald summation method that are available in the Material Studio simulation package. The aim of this work is to investigate the solubility and intermolecular interactions (i.e., hydrogen bonding) of water with mefenamic acid. The solubility of mefenamic acid in water is lower than the ideal solubility. The results of the simulation show that the density, diffusion coefficient and radial distribution functions (RDFs) calculated for water are comparable with the literature. Analysis of radial distribution functions in the binary system of mefenamic acid/water shows low hydrogen bond formation between mefenamic acid and water molecules as well as diffusivity. In addition, the strength of the interaction is much lower than that of hydrogen bonds formed between molecules of water. These findings suggest low diffusivity and poor solubility characteristic of mefenamic acid in water

Application of particle-mesh Ewald summation to ONIOM theory

We extended a particle mesh Ewald (PME) summation method to the ONIOM (our Own N-layered Integrated molecular Orbitals and molecular Mechanics) scheme (PME-ONIOM) to validate the simulation in solution. This took the form of a nonadiabatic ab initio molecular dynamics (MD) simulation in which the Zhu-Nakamura trajectory surface hopping (ZN-TSH) method was performed for the photoisomerization of a (Z)-penta-2,4-dieniminium cation (protonated Schiff base, PSB3) electronically excited to the S1 state in a methanol solution. We also calculated a nonadiabatic ab initio MD simulation with only minimum image convention (MI-ONIOM). The lifetime determined by PME-ONIOM-MD was 3.483ps. The MI-ONIOM-MD lifetime of 0.4642ps was much shorter than those of PME-ONIOM-MD and the experimentally determined excited state lifetime. The difference eminently illustrated the accurate treatment of the long-range solvation effect, which destines the electronically excited PSB3 for staying in S1 at the pico-second or the femto-second time scale.

Computation of the singular and regularized image systems for doubly-periodic Stokes flow in the presence of a wall

A method is presented for the efficient computation of Stokes flow due to doubly-periodic arrays of regularized forces and point forces in 3D near a wall. Our approach is based on a variation of Ewald's summation method and the method of images for regularized and singular Stokeslets. The slowly convergent series resulting from the direct summation of the doubly-periodic array of Stokeslets is recast into the sum of two series, one in real space and one in Fourier space, both having partial sums decaying in a Gaussian manner. The formulation of our method guarantees that the velocity on the wall is analytically zero independent of the number of summands used. We also introduce an empirical approach to compute the ‘optimal’ number of summands in real space and Fourier space corresponding to a given splitting parameter. Finally, we report computational run times on a CPU and a GPU, and give two numerical examples to illustrate the method.

Application of isotropic periodic sum method for 4-pentyl-4′-cyanobiphenyl liquid crystal

In future large-scale molecular dynamics (MD) simulations that will use parallel computing, the isotropic periodic sum (IPS) method is expected to effectively reduce the cost of interaction calculations while maintaining adequate accuracy. To assess the accuracy of this method in estimating low-charge-density polymer systems, we performed atomistic MD simulations of the bulk state of liquid crystal systems based on 4-pentyl-4′-cyanobiphenyl (5CB). In conditions of 270 K ≤ T ≤ 320 K and a normal pressure, the temperature dependence of the density, potential energy and order parameter was estimated using the IPS and Ewald sum method. The results of the IPS method and Ewald sum were consistent within the range of error. In conditions close to the phase transition point, however, the averaged values of potential energy and order parameter had a small difference. We concluded that the fundamental physical properties for the bulk state of 5CB systems are determined reasonably by using the IPS method, at least in conditions that are not close to the phase transition point.

Efficient Handling of Gaussian Charge Distributions: An Application to Polarizable Molecular Models.

We present a mesh-based Ewald summation method that is suitable for the calculation of the electrostatic interaction between Gaussian charge distributions, instead of point charges. As an application, we implemented the method in the Gromacs simulation package and tested it with a polarizable water model, showing that the interaction between Gaussian charge distributions can be computed with a small (10%) additional computational cost with respect to the point charge case. In addition, since the performance of polarizable models is strongly influenced by the number of iterations required for the self-consistent field solution, we tested also the Always Stable Predictor-Corrector (ASPC) method of Kolafa (Kolafa, J. J. Comp. Chem. 2003, 25, 335) as an alternative to the steepest descent (SD) based algorithm with predictor implemented in the Gromacs, and found that it speeds up the integration of the equations of motion by a factor of 1.6-2.0, depending on the target model.

Acceleration of Ab Initio QM/MM Calculations under Periodic Boundary Conditions by Multiscale and Multiple Time Step Approaches

Development of multiscale ab initio quantum mechanical and molecular mechanical (AI-QM/MM) method for periodic boundary molecular dynamics (MD) simulations and their acceleration by multiple time step approach are described. The developed method achieves accuracy and efficiency by integrating the AI-QM/MM level of theory and the previously developed semiempirical (SE) QM/MM-Ewald sum method [J. Chem. Theory Comput. 2005, 1, 2] extended to the smooth particle-mesh Ewald (PME) summation method. In the developed methods, the total energy of the simulated system is evaluated at the SE-QM/MM-PME level of theory to include long-range QM/MM electrostatic interactions, which is then corrected on the fly using the AI-QM/MM level of theory within the real space cutoff. The resulting energy expression enables decomposition of total forces applied to each atom into forces determined at the low-level SE-QM/MM method and correction forces at the AI-QM/MM level, to integrate the system using the reversible reference system propagator algorithm. The resulting method achieves a substantial speed-up of the entire calculation by minimizing the number of time-consuming energy and gradient evaluations at the AI-QM/MM level. Test calculations show that the developed multiple time step AI-QM/MM method yields MD trajectories and potential of mean force profiles comparable to single time step QM/MM results. The developed method, together with message passing interface (MPI) parallelization, accelerates the present AI-QM/MM MD simulations about 30-fold relative to the speed of single-core AI-QM/MM simulations for the molecular systems tested in the present work, making the method less than one order slower than the SE-QM/MM methods under periodic boundary conditions.

A determination of liquid–vapour interfacial properties for methanol using a linear-combination-based isotropic periodic sum

We performed molecular dynamics simulations of the methanol liquid–vapour interfacial systems for a wide temperature range of 200–350 K. The linear-combination-based isotropic periodic sum (LIPS) method was used for estimating Coulombic interactions of the systems. The temperature dependence of the liquid and vapour phase densities, surface thickness and surface tension using the LIPS method was almost equal to that of the Ewald sum method. The temperature dependence of the electrostatic potential profile using the LIPS method was very similar to that of the Ewald sum method. We conclude that the LIPS method provides a reasonable accuracy in determining the liquid–vapour interfacial properties of polar molecules for many temperature conditions.

Petascale molecular dynamics simulation using the fast multipole method on K computer

In this paper, we report all-atom simulations of molecular crowding — a result from the full node simulation on the “K computer”, which is a 10-PFLOPS supercomputer in Japan. The capability of this machine enables us to perform simulation of crowded cellular environments, which are more realistic compared to conventional MD simulations where proteins are simulated in isolation. Living cells are “crowded” because macromolecules comprise ∼30% of their molecular weight. Recently, the effects of crowded cellular environments on protein stability have been revealed through in-cell NMR spectroscopy. To measure the performance of the “K computer”, we performed all-atom classical molecular dynamics simulations of two systems: target proteins in a solvent, and target proteins in an environment of molecular crowders that mimic the conditions of a living cell. Using the full system, we achieved 4.4 PFLOPS during a 520 million-atom simulation with cutoff of 28 Å. Furthermore, we discuss the performance and scaling of fast multipole methods for molecular dynamics simulations on the “K computer”, as well as comparisons with Ewald summation methods.

Non-Uniform FFT and Its Applications in Particle Simulations

Ewald summation method, based on Non-Uniform FFTs (ENUF) to compute the electrostatic interactions and forces, is implemented in two different particle simulation schemes to model molecular and soft matter, in classical all-atom Molecular Dynamics and in Dissipative Particle Dynamics for coarse-grained particles. The method combines the traditional Ewald method with a non-uniform fast Fourier transform library (NFFT), making it highly efficient. It scales linearly with the number of particles as , while being both robust and accurate. It conserves both energy and the momentum to float point accuracy. As demonstrated here, it is straight- forward to implement the method in existing computer simulation codes to treat the electrostatic interactions either between point-charges or charge distributions. It should be an attractive alternative to mesh-based Ewald methods.

Coarse-grained simulations of the salt dependence of the radius of gyration of polyelectrolytes as models for biomolecules in aqueous solution

The salt dependent radius of gyration of a polyelectrolyte in aqueous solution is calculated in an environment where the polyelectrolyte is surrounded by a permeable membrane that exchanges only solvent particles with the bulk. We obtain additionally the scaling exponent of the gyration radius as a function of the polymerization degree, and find that the polyelectrolyte retains a stretched conformation during the condensation and re-expansion process, indicating that these effects are of an electrostatic nature. The solvent quality is also shown to affect the polyelectrolyte conformation, especially for the poor solvent case. These results are obtained using a hybridized Monte Carlo technique with the coarse-grained, dissipative particle dynamics method with fluctuating number of solvent particles. The full range of the electrostatic interactions is included in the simulations, using the Ewald sum method, and the counterions and solvent molecules are included explicitly. In the complex systems mentioned above, the electrostatic interactions and the solvent quality play a key role in understanding phenomena that do not occur in uncharged systems. Our results are compared and validated with the behavior of some biomolecules under similar environments.

Dissipative particle dynamics simulations of weak polyelectrolyte adsorption on charged and neutral surfaces as a function of the degree of ionization

The influence of the chain degree of ionization on the adsorption of weak polyelectrolytes on neutral and on oppositely and likely charged surfaces is investigated for the first time, by means of Monte Carlo simulations with the mesoscopic interaction model known as dissipative particle dynamics. The electrostatic interactions are calculated using the three-dimensional Ewald sum method, with an appropriate modification for confined systems. Effective wall forces confine the linear polyelectrolytes, and electric charges on the surfaces are included. The solvent is included explicitly also and it is modeled as an athermal solvent for the polyelectrolytes. The number of solvent particles is allowed to fluctuate. The results show that the polyelectrolytes adsorb both onto neutral and charged surfaces, with the adsorption regulated by the chain degree of ionization, being larger at lower ionization degrees, where polyelectrolytes are less charged. Furthermore, polyelectrolyte adsorption is strongly modulated by the counterions screening of surface charge. These findings are supported with predictions of adsorption isotherms with varying ionization degree. We obtain also the surface force mediated by adsorbed polyelectrolytes, which is calculated for the first time as a function of ionization degree. The adsorption and surface force isotherms obtained for weak polyelectrolytes are found to reproduce main trends in experiments, whenever those results are available, and provide additional insight into the role played by the competitive adsorption of the counterions and polyelectrolytes on the surfaces.

Optimization of a Brownian-dynamics algorithm for semidilute polymer solutions

Simulating the static and dynamic properties of semidilute polymer solutions with Brownian dynamics (BD) requires the computation of a large system of polymer chains coupled to one another through excluded-volume and hydrodynamic interactions. In the presence of periodic boundary conditions, long-ranged hydrodynamic interactions are frequently summed with the Ewald summation technique. By performing detailed simulations that shed light on the influence of several tuning parameters involved both in the Ewald summation method, and in the efficient treatment of Brownian forces, we develop a BD algorithm in which the computational cost scales as O(N(1.8)), where N is the number of monomers in the simulation box. We show that Beenakker's original implementation of the Ewald sum, which is only valid for systems without bead overlap, can be modified so that θ solutions can be simulated by switching off excluded-volume interactions. A comparison of the predictions of the radius of gyration, the end-to-end vector, and the self-diffusion coefficient by BD, at a range of concentrations, with the hybrid lattice Boltzmann-molecular dynamics (LB-MD) method shows excellent agreement between the two methods. In contrast to the situation for dilute solutions, the LB-MD method is shown to be significantly more computationally efficient than the current implementation of BD for simulating semidilute solutions. We argue, however, that further optimizations should be possible.

Simplistic Coulomb Forces in Molecular Dynamics: Comparing the Wolf and Shifted-Force Approximations

In this paper we compare the Wolf method to the shifted forces (SF) method for efficient computer simulation of bulk systems with Coulomb forces, taking results from the Ewald summation and particle mesh Ewald methods as representing the true behavior. We find that for the Hansen-McDonald molten salt model the SF approximation overall reproduces the structural and dynamical properties as accurately as does the Wolf method. It is shown that the optimal Wolf damping parameter depends on the property in focus and that neither the potential energy nor the radial distribution function are useful measures for the convergence of the Wolf method to the Ewald summation method. The SF approximation is also tested for the SPC/Fw model of liquid water at room temperature, showing good agreement with both the Wolf and the particle mesh Ewald methods; this confirms previous findings [Fennell, C. J.; Gezelter, J. D. J. Chem. Phys.2006, 124, 234104]. Besides its conceptual simplicity, the SF approximation implies a speed-up of a factor of 2-3 compared to the Wolf method. We conclude that for the systems studied, whenever the Wolf method gives accurate results, it may be replaced by the simpler and faster SF method.

Confining dyon gas with finite-volume effects under control

As an approach to describe the long-range properties of non-Abelian gauge theories at non-zero temperature T < T_c, we consider a non-interacting ensemble of dyons (magnetic monopoles) with non-trivial holonomy. We show analytically, that the quark-antiquark free energy from the Polyakov loop correlator grows linearly with the distance, and how the string tension scales with the dyon density. In numerical treatments, the long-range tails of the dyon fields cause severe finite-volume effects. Therefore, we demonstrate the application of Ewald's summation method to this system. Finite-volume effects are shown to be under control, which is a crucial requirement for numerical studies of interacting dyon ensembles.

Physical meaning of the Ewald sum method

The electrostatic potential and energy of point charges in a real crystal, in the presence of thermal vibrations, is obtained as a special case of the Fourier method. Incorporating the role of thermal vibrations in electrostatic energy calculations leads to the physical meaning of the Ewald sum method. The Ewald summation method determines the electrostatic potential and energy of point charges in a crystal at an equivalent temperature that is obtained from the width of the Gaussian and not at 0 K. For values of the width of the Gaussian commonly recommended for computational convenience, the equivalent temperatures exceed 10,000 K.

Collective electronic excitation coupling between planar optical lattices using Ewald's method

Using Ewald's summation method, we investigate collective electronic excitations (excitons) of ultracold atoms in parallel planar optical lattices including long-range interactions. The exciton dispersion relation can then be suitably rewritten and efficiently calculated for long-range resonance dipole–dipole interactions. Such in-plane excitons resonantly couple for two identical optical lattices, with an energy transfer strength decreasing exponentially with the distance between the lattices. This allows a restriction of the transfer to neighbouring planes and gives rise to excitons delocalized between the lattices. In general, equivalent results will hold for any planar system containing lattice layers of optically active and dipolar materials.

Assessing group-based cutoffs and the Ewald method for electrostatic interactions in clusters and in saturated, superheated, and supersaturated vapor phases of dipolar molecules

Monte Carlo simulations in the canonical, isobaric-isothermal, grand canonical, and Gibbs ensembles were used to assess whether the computationally expensive Ewald summation method for the computation of the first-order electrostatic energy can be replaced with a simpler truncation approach for accurate simulations of the saturated, superheated, and supersaturated vapor phases of dipolar and hydrogen-bonding molecules. Rotationally averaged hydrogen fluoride dimer and trimer energies, thermophysical properties and aggregation in the superheated vapor phase of hydrogen fluoride, nucleation free energy barriers for water, and the vapor–liquid coexistence properties of hydrogen fluoride and water were investigated over a wide range of state points. We find that for densities not too close to the critical density, results obtained from simulations using a spherical potential truncation based on neutral groups (molecules or fragments) for the Coulomb interactions are statistically identical to those obtained using the Ewald summation method. Use of the simpler spherical truncation results in a significant reduction of the computational effort for simulations employing molecular mechanics force fields and also allows for straightforward implementation of many-body expansion methods to compute the potential energy from electronic structure calculations of subsystems of the entire vapor-phase system.